Bayesian Hyperbolic Multidimensional Scaling
提出一种贝叶斯方法，将高维数据映射到双曲空间进行降维，适用于树状结构数据（如文本或基因数据），并通过近似方法将计算复杂度从O(n²)降至O(n)。
Multidimensional scaling (MDS) is a widely used approach for representing high-dimensional, dependent data. MDS works by assigning each observation a location on a low-dimensional geometric manifold, with distance on the manifold representing similarity. We propose a Bayesian approach to multidimensional scaling when the low-dimensional manifold is hyperbolic. Using hyperbolic space facilitates representing tree-like structures common in many settings (e.g., text or genetic data with hierarchical structure). A Bayesian approach provides regularization that minimizes the impact of measurement error in the observed data and assesses uncertainty. We also propose a case-control likelihood approximation that allows for efficient sampling from the posterior distribution in larger data settings, reducing computational complexity from approximately O(n2) to O(n). We evaluate the proposed method against state-of-the-art alternatives using simulations, canonical reference datasets, Indian village network data, and human gene expression data. Code to reproduce the result in the article is available at https://github.com/peterliu599/BHMDS. Supplementary materials for this article are available online.